Question

Cos^4 A - Sin^4 + 1 = 2cos^2 A​

Answer 1

Step-by-step explanation:

Given :

• To prove

$\mathsf{ {\cos(a) }^{4} - { \sin(a) }^{4} + 1 = 2 { \cos(a) }^{2}}$

• Proof

Solving LHS

$\mathsf{ {\cos(a) }^{4} - { \sin(a) }^{4} + 1}$

$\mathsf{\implies {{(\cos(a) }^{2})}^{2} - { \sin(a) }^{4}+1}$

Now By equation

$\boxed{\bigstar \:{\sin{A}}^{2} - {\cos{A}}^{2} = 1}$

$\mathsf{\implies {{(\sin(a) }^{2} - 1 )}^{2} - {\sin(a)}^{4}+1}$

Now opening the brackets we have

$\mathsf{\implies {\sin(a) }^{4} + 1 -2 { \sin(a)}^{2} - {\sin(a)}^{4} +1}$

$\mathsf{\implies 2-2{\sin(a)}^{2}}$

$\mathsf{\implies 2(1-{\sin(a)}^{2})}$

Now similarly by above equation we have

$\mathsf{\implies 2{\cos(a)}^{2}}$

LHS = RHS

Hence proved :